The true statement for all invertible matrices A and B are
1. (In−A)(In+A)=In−A².
2. (AB)⁻¹=A⁻¹B⁻¹
4. A⁷ is invertible.
The given statement is true for all invertible matrices A. To prove this statement, we can expand the left-hand side of the equation as follows:
(In−A)(In+A) = In(In) + In(A) − A(In) − A(A)
= In² + InA − AIn − A²
= In + InA − AIn − A²
= In − A²
Therefore, we have shown that (In−A)(In+A)=In−A2 is true for all invertible matrices A.
The statement is true for all invertible matrices A and B. To prove this statement, we can use the definition of the inverse of a matrix. The inverse of a matrix A is a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I, where I is the identity matrix. Using this definition, we can show that:
(AB)(A⁻¹B⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = AA⁻¹ = I
(B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹IB = BB⁻¹ = I
Therefore, we have shown that (AB)⁻¹ = A⁻¹B⁻¹ is true for all invertible matrices A and B.
The statement is false in general. For instance, consider the matrices A = [1 0] and B = [−1 0]. Both A and B are invertible matrices, but A + B = [0 0] which is not invertible as it is not a full rank matrix.
The statement is true for all invertible matrices A. To prove this statement, we can use the fact that the product of invertible matrices is also invertible. Since A is invertible, we can write:
A⁷ = AAAA...A
= A⁶A
= (A⁻¹)⁻¹A⁶A
= (A⁻¹A)⁻¹A⁶A
= IA⁶A
= A⁶
We can repeat this process until we get A⁷ = (A⁻¹)⁻¹. Thus, A⁷ is invertible for all invertible matrices A.
The statement is false in general. To show this, we can use a counterexample. Let A = [1 0] and B = [0 −1]. Then,
(A + B)² = [1 −1][1 −1]
= [0 0]
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jerome haw 1,040 songs downloaded on his spotify account and 30% of the songs are country songs. How many of the songs are not country
Question 19 (2 points)
According to research conducted by the Department of Education, 80% of college
students took a mathematics course as part of their general education requirements.
If ten college students are selected at random, what is the probability at least one of
the ten has not taken a mathematics course?
0.0800
0.1073
0.7927
0.8000
Rounding to four decimal places, the answer is 0.8926. So the probability that at least one of the ten college students has not taken a mathematics course is approximately 0.8926.
what is probability?
Probability is a measure of the likelihood or chance that a particular event will occur. It is usually expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
The probability that a college student has taken a mathematics course is 0.80. Therefore, the probability that a student has not taken a mathematics course is
1 - 0.80 = 0.20.
The probability that at least one of ten college students has not taken a mathematics course can be found using the complement rule. That is, the probability that at least one student has not taken a mathematics course is equal to one minus the probability that all ten students have taken a mathematics course.
The probability that all ten students have taken a mathematics course is:
0.80¹⁰ = 0.1074
Therefore, the probability that at least one student has not taken a mathematics course is:
1 - 0.1074 = 0.8926
Therefore, Rounding to four decimal places, the answer is 0.8926. So the probability that at least one of the ten college students has not taken a mathematics course is approximately 0.8926.
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If a ll b, find the value of x.
Answer:
x = 18
Step-by-step explanation:
Alternate exterior angles are congruent. Set the equations equal to each other and solve for x.
7x + 11 = 10x - 43 Subtract 7 x from both sides
7x - 7x + 11 = 10x - 7x - 43
11 = 3x - 43 Add 3 to both sides
11 + 43 = 3x -43 + 43
54 = 3x Divide both sides by 3
[tex]\frac{54}{3}[/tex] = [tex]\frac{3x}{3}[/tex]
18 = x
Helping in the name of Jesus.
Which of the following is NOT a procedure for determining whether it is reasonable to assume that sample data are from a normally distributed population? Choose the correct answer below /08/19 1:59pm 3/15/19 1:59pm O A. Identifying outliers O B. Checking that the probability of an event is 0.05 or less OC. Visual inspection of a histogram to see if it is roughly bell-shaped OD. Constructing a graph called a normal quantile plot 1/29/19 1:59pm
Therefore , the solution of the given problem of probability comes out to be (B) Verifying that an event's chance is 0.05 or less is the right response.
What is probability exactly?The primary goal of a procedure's criteria-based methods is to calculate the probability that a statement is true or that a specific occurrence will occur. Any number range from 0 to 1, where 0 usually represents the likelihood of something happening and 1 typically represents an amount of confidence, can be used to represent chance. A probability illustration displays the possibility that a specific event will take place.
Here,
Several techniques can be used to determine whether it is reasonable to infer that sample data come from a population with a normally distributed population, including:
A. Recognizing anomalies
B. Verifying that an event's probability is 0.05 or lower C.
Examining a histogram visually to see if it approximately resembles a bell shape
D. Creating a normal quantile plot, a type of graph.
Option B is invalid because it doesn't reveal anything about how the sample data are distributed.
The threshold for statistical significance is the chance of an event being 0.05 or less, but it has no bearing on the distribution's shape.
As a result, (B) Verifying that an event's chance is 0.05 or less is the right response.
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PLEASE HELPP DUE TODAY!!!!!
Calculate the value of c in the triangle below.
Answer:
82
Step-by-step explanation:
what is the ordered pair to this question using substitution? y = −2x − 7
2y − x = 1
Answer: (-3, -1)
Step-by-step explanation:
First, we can use the first equation to solve for one variable in terms of the other.
y = -2x - 7
Next, we can substitute this expression for y into the second equation:
2y - x = 1
2(-2x - 7) - x = 1
Simplifying:
-4x - 14 - x = 1
-5x - 14 = 1
-5x = 15
x = -3
Now that we have the value of x, we can substitute it back into either of the original equations to find y. Using the first equation:
y = -2(-3) - 7
y = 6 - 7
y = -1
Therefore, the ordered pair that satisfies both equations is (-3, -1).
Answer: (-3,-1)
Explanation Step By Step
Which expression represents the distance
between point G and point H?
|-12|16| |-12|+|-9|
1-9|-|-6|
|-12|+|6|
-15
H(-9,6)
G(-9,-12)
15+y
0
-15-
15
Answer:
Step-by-step explanation:
2
What is the percent of change from 100 to 85?
Hi!
The percent change is 15%.
Hope this helps!
~~~PicklePoppers~~~
19
The diagram shows a sector OBC of a circle with centre O and radius (6 + x) cm.
B
A
es 50°
6 cm
DX
find the value of x.
Give your answer correct to 3 significant figures.
x cm
A is the point on OB and D is the point on OC such that OA = OD = 6 cm
Angle BOC= 50°
Given that
Diagram N
accurately
the perimeter of sector OBC= 2 x the perimeter of triangle OAD
The value of the variable, x, obtained using the formula for the perimeter of a sector of a circle is; x = 6 cm
What is a sector of a circle?A sector of a circle is the region of the circle that has two radii of the circle and the arc between the two radii as the boundary of the region.
The perimeter of a sector of a circle can be obtained using the formula;
Perimeter = 2·r + (θ/360°) × (2·π·r)
The radius of sector OAD, r = 6 cm
Therefore;
The perimeter of sector OAD = 2 × 6 + (50/360) × 2 × π × 6 = 12 + 5·π/3
The perimeter of sector OBC = 2 × The perimeter of sector OAD
Therefore;
The perimeter of sector OBC = 2 × (12 + 5·π/3) = 24 + 10·π/3
Which indicates;
The perimeter of sector OBC = 2 × (6 + x) + (50/360) × 2 × π × (6 + x)
2 × (6 + x) + (50/360) × 2 × π × (6 + x) = (5·π·x + 30·π + 36·x + 216)/18
24 + 10·π/3 = (5·π·x + 30·π + 36·x + 216)/18
432 + 60·π = 5·π·x + 30·π + 36·x + 216
432 - 216 + 60·π - 30·π = x·(5·π + 36)
216 + 30·π = x·(5·π + 36)
6 × (5·π + 36 ) = x·(5·π + 36)
x·(5·π + 36) = 6 × (5·π + 36 )
x = 6 × (5·π + 36 )/(5·π + 36 ) = 6
x = 6 cm
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A human head can be approximated as a sphere with a circumference of 60
centimeters. What is the approximate volume of a human head, rounded to the nearest 1,000
cubic centimeters?(GEOMETRY HELP)
The approximate volume of a human head is 4000 cubic centimeter.
What is the approximate volume of a human head?A sphere is simply a geometrical object that is a three-dimensional analogue to a two-dimensional circle.
The circumference of a sphere is given by the formula:
C = 2πr
Where r is the radius of the sphere. In this case, we are given that the circumference of the sphere (which approximates the human head) is 60 cm, so:
60 cm = 2πr
Solve for r
r = 60/2π
r = 30/π
The volume of a sphere is given by the formula:
V = (4/3)πr³.
Substituting the value we found for r, we get:
V = (4/3) × π × ( 30/π )³
V = (4/3) × π × ( 30/π )³
V = 3647.566 cm³
V = 4000 cm³
Therefore, the volume to the nearest 1,000 cubic cm is 4000 cm³.
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You determine the percent abundance of
each length of nail and record it in the data
table below.
Sample
Type
Short nail
Medium nail
Long nail
Number Abundance
of Nails
(%)
67
18
10
70.5
19.0
10.5
Nail Length
(cm)
2.5
5.0
7.5
What is the weighted average length, in cm,
of a nail from the carpenter's box?
The weighted average length of a nail from the carpenter's box is 3.5 centimeters.
How to calculate the weighted average length?Different from calculating the average, the weighted average implies considering the frequency or abundance percentage. Now, to calculate the average weighted we will need to multiply the length of each type of nail by the abundance and finally, we will need to add the results obtained. The process is shown below:
Short nail: 2.5 cm x 70.5%= 1.7625 cm
Medium nail: 5.0 cm x (19% = 0.95 cm
Long nail: 7.5 cm x 10.5% = 0.7875 cm
1.7625 cm + 0.95 cm + 0.7875 cm = 3.5 cm
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a function ___ specifies the return data type, name of the function, and the parameter variable(s).
A function declaration specifies the return data type, name of the function, and the parameter variable(s).
In programming, a function declaration is a statement that specifies the characteristics of a function. It includes the name of the function, the return data type (if any), and the parameter variable(s) (if any) that the function expects to receive as input. The declaration is used to inform the compiler or interpreter about the existence and behavior of the function, so that it can be called from other parts of the program. The function's actual implementation or definition is typically written separately from the declaration. By separating the declaration and implementation of a function, programs can be more modular and easier to maintain.
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P, Q, R, S, T and U are different digits.
PQR + STU = 407
Step-by-step explanation:
There are many possible solutions to this problem, but one possible set of values for P, Q, R, S, T, and U is:
P = 2
Q = 5
R = 1
S = 8
T = 9
U = 9
With these values, we have:
PQR = 251
STU = 156
And the sum of PQR and STU is indeed 407.
8. What is the area of sector EFG? Express the answer
in terms of л.
10
E
128⁰
G
The area of a sector is a region bounded by two radii of a circle and an arc between is (320/9) π sq. units.
What is radius?Radius is a straight line segment that connects the center of a circle or sphere to any point on its circumference or surface, respectively. In other words, it is the distance from the center of the circle or sphere to any point on its edge or surface.
According to question:The given sector is EFG = 128°.
The area of a sector is a region bounded by two radii of a circle and an arc between them.
A = (θ/360) × πr²
Where:
A is the area of the sectorθ is the central angle of the sector in degreesr is the radius of the circleThe fraction (θ/360) represents the fraction of the entire circle that the sector occupies. Multiplying this fraction by the total area of the circle (πr²) gives the area of the sector.
Use the formula for the area of a sector with the central angle in degrees:
A = (n/360) × πr²
From the given figure, n = 128 and r = 10
A = (128/360) × π(10)²
= (320/9) π sq. units.
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What is the smallest possible integer for which 18% of that integer is greater than 3.5 ?
A14
B 16
C 18
D 20
E 22
Answer:
D 20
Step-by-step explanation:
Let's call the integer we're looking for "x". We know that 18% of x is greater than 3.5, so we can write the inequality:
0.18x > 3.5
To solve for x, we can divide both sides by 0.18:
x > 3.5 ÷ 0.18
x > 19.44
We want the smallest possible integer that satisfies this inequality, which is 20. So the answer is D) 20.
Which graph matches the function given:
The graph that matches the piecewise function, f(x) = √(x + 5), if x < -2, f(x) = |x + 1| if -2 ≤ x ≤ 2, and f(x) = (x - 2)² if x > 2 is the graph in the third option.
What is a piecewise function?A piecewise function is a function is a function that consists of two or more subfunctions each of which are applied, based on the specific interval of the input variable.
The intervals of the piecewise function are;
f(x) = √(x + 5) if x < -2
f(x) = |x + 1| -2 ≤ x ≤ 2
f(x) = (x - 2)² if x > 2
The graph of the piecewise function is a three piece graph which consists of the graph of f(x) = √(x + 5), for x values less than -2, f(x) = |x + 1|, for x-values in the interval -2 ≤ x ≤ 2 and the graph of f(x) = (x - 2)²
The <-2, symbol indicates the presence of an open circle in the graph of f(x) = √(x + 5) at x = -2
The interval -2 ≤ x ≤ 2 for the function f(x) = |x + 1| indicates that the graph of f(x) = |x + 1| in the interval -2 ≤ x ≤ 2, consists of closed circles at x = -2 and x = 2.
The interval, x > 2, for the function, f(x) = (x - 2)², indicates that the presence of an open circle in the graph of f(x) = (x - 2)² at x = 2.
The correct option for the graph of the piecewise function is therefore the third option.
Please find the attached the graph of the piecewise function created with MS Excel
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CAN SOMEONE HELP WITH THIS QUESTION?✨
Step-by-step explanation:
my previous answer assumed we need a kind of average change rate.
but it seems you need the immediate change rate ?
or what ? the question does not specify.
I would appreciate if you would leave a comment, when you find that my answer is seemingly not matching some expected answers.
so, let's try now the immediate change rate (as I have no information to go - I have also no information about what you are currently learning in your class, and if you do already derivatives or not, so, this might be now above your learning level ...) :
again, the area between the inner circle and the outer square is
A = As - Ac = s² - pi×r²
where s is the square's side length, and r is the radius of the circle.
the immediate change rate of that area is the sum of first derivatives of A based on both used variables multiplied by their individual change rates :
dA/dt = dAs/ds × ds/dt - dAc/dr × dr/dt
and this is then calculated for the given data point (s = 16, r = 3) and the individual change rates of the variables (ds/dt = 2 meters/minute, dr/dt = 1 meter/minute).
so, we have
dA/dt = 2s × 2 - 2×pi×r × 1
as (s²)' = 2s, (pi×r²)' = 2×pi×r
dA(16, 3)/dt = 2×16×2 - 2×pi×3×1 = 64 - 6pi =
= 45.15044408... meters²/minute ≈
≈ 45.15 meters²/minute
so, you see, for a curved function (not a line) there has to be a difference between the average change rate between 2 points and the immediate change rate directly at a point.
I hope this works now for you.
and here now the original answer for an average change rate :
right now the radius is 3 m, and the square side lengths are 16 m.
1 minute ago the radius was 2 m, and the square side lengths were 14 m.
the area between the circle and the square is always the area of the square minus the area of the circle.
these areas are right now
circle : pi×3² = 9pi m²
square : 16² = 256 m²
area between = 256 - 9pi = 227.7256661... m²
these areas were one minute ago
circle : pi×2² = 4pi m²
square : 14² = 196 m²
area between = 196 - 4pi = 183.4336294... m²
the change rate of the area between the circle and the square is
227.7256661... - 183.4336294... = 44.29203673... m² per minute
Give the interval(s) on which the function is continuous.
g(t) = 1/√16-t^2
The function g(t) is defined as:
g(t) = 1/√(16-t^2)
The function is continuous for all values of t that satisfy the following conditions:
The denominator is non-zero:
The denominator of the function is √(16-t^2). Therefore, the function is undefined when 16-t^2 < 0, or when t is outside the interval [-4,4].
There are no vertical asymptotes:
The function does not have any vertical asymptotes, because the denominator is always positive.
Thus, the function g(t) is continuous on the interval [-4,4].
Find the volume of the solid generated by revolving the region enclosed by the following curves: a. y=4-2x², y = 0, x = 0, y = 2 through 360° about the y-axis. b. x = √y+9, x = 0, y =1 through 360° about the y-axis.
The volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
How to find the volume?To find the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis, we use the formula:
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to find the equation of the curve that is generated when we rotate y = 4 - 2x² around the y-axis. To do this, we use the formula for the equation of a curve generated by revolving y = f(x) around the y-axis, which is:
x² + y² = r²
where r is the distance from the y-axis to the curve at any point (x, y).
Substituting y = 4 - 2x² into this formula, we get:
x² + (4 - 2x²) = r²
Simplifying, we get:
r² = 4 - x²
Therefore, the radius of the circular cross-sections perpendicular to the y-axis is given by:
r = √(4 - x²)
Now, we can integrate πr²dy from y = 0 to y = 2:
V = ∫[0,2] π(√(4 - x²))²dy
V = ∫[0,2] π(4 - x²)dy
V = π∫[0,2] (4y - y²)dy
V = π(2y² - (1/3)y³)∣[0,2]
V = π(8 - (8/3))
V = (16/3)π
Therefore, the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
b. To find the volume of the solid generated by revolving the region enclosed by the curves x = √y+9, x = 0, and y = 1 through 360° about the y-axis, we use the same formula as in part (a):
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to solve the equation x = √y+9 for y in terms of x:
x = √y+9
x² = y + 9
y = x² - 9
Next, we need to find the equation of the curve that is generated when we rotate y = x² - 9 around the y-axis. Using the same formula as in part (a), we get:
r = x
Now, we can integrate πr²dy from y = 1 to y = 10 (since x = 0 when y = 1 and x = 3 when y = 10):
V = ∫[1,10] π(x²)²dy
V = π∫[1,10] x⁴dy
V = π(1/5)x⁵∣[0,3]
V = π(243/5)
Therefore, the volume of the solid generated by revolving the region.
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C is 2 units closer to B than it is to A, if A = 5, B=15
Answer:
the coordinate of C is 10.5, and the distance between A and C is 10.5 - 5 = 5.5, and the distance between B and C is 15 - 10.5 = 4.5.
Step-by-step explanation:
If C is 2 units closer to B than it is to A, then we can find the distance between A and C and between B and C by using the distance formula:
Distance between two points (x1, y1) and (x2, y2) = sqrt[(x2 - x1)^2 + (y2 - y1)^2]
Let x be the coordinate of C. Then, the distance between A and C is x - 5, and the distance between B and C is 15 - x. We know that the distance between B and C is 2 units greater than the distance between A and C, so we can set up an equation:
15 - x = 2 + (x - 5)
Simplifying and solving for x, we get:
15 - x = 2 + x - 5
18 - x = x - 3
2x = 21
x = 10.5
Therefore, the coordinate of C is 10.5, and the distance between A and C is 10.5 - 5 = 5.5, and the distance between B and C is 15 - 10.5 = 4.5.
The mean weight of 4 parcels is 8.5kg. Three of them weighed 7.7 kg, 7.6 kg and 8.2 kg.
What is the weight of the fourth parce1?
Answer:
Weight of the fourth parcel will be 10.5 kgStep-by-step explanation:
Weight of first parcal = 7.7 kg Weight of second parcel = 7.6 kgWeight of third parcel = 8.2 kg Mean Weight = 8.5 kgLet weight of fourth parcel be x
Mean = Sum of all values/total number of values.
8.5 = 7.7 + 7.6 + 8.2 + x/4
8.5 = 23.5 + x/4
8.5 × 4 = 23.5 + x
34 = 23.5 + x
34 - 23.5 = x
10.5 = x
Therefore, weight of the fourth parcel will be 10.5 kg
What is the end behavior of the polynomial function?
Answer: D. As x → -∞, y → -∞.
Step-by-step explanation:
The graph shows the function approaching negative infinity on the x-axis (left side). When the x-axis is decreasing, the y-axis is also decreasing towards negative infinity.
HELP MY TEST IS DUE TONIGHT
find the area of the trapezoid
Show work:
Answer:
A = 32
Step-by-step explanation:
A= (a+b divided by 2) mulitiplied by height
a = 6
b = 2+6+2 = 10
h = 4
a+b = 6 + 10= 16
16 divided by 2 = 8
8 x 4 = 32
Therefore, the area is 32.
Question 41 (Essay Worth 10 points)
(06.06 HC)
Use specific examples from the history you have learned to compose a well-developed paragraph on the following:
How has the rise of democracy impacted ONE of the following regions: Africa, Asia, or the Caribbean?
Answer:
See below, please.
Step-by-step explanation:
The rise of democracy has had a significant impact on the African continent, with many countries transitioning from authoritarian rule to democratic governance since the 1990s. For example, South Africa emerged from decades of apartheid and transitioned to democracy in 1994, with the election of Nelson Mandela as the country's first black president. Similarly, in Ghana, the return to multiparty democracy in 1992 after years of military rule led to the establishment of a more stable political system.
The impact of democracy has not been uniform across Africa, with some countries experiencing significant challenges in their democratic transitions. For instance, in Zimbabwe, President Robert Mugabe, who came to power after independence in 1980, held on to power for over three decades through rigged elections and suppression of political opposition. It wasn't until a military coup in 2017 that he was forced to step down.
Despite the challenges, democracy has brought about some significant improvements on the African continent. With democratic governance, there has been a greater emphasis on human rights, the rule of law, and good governance. Free and fair elections have become the norm in many African countries, and citizens have become more involved in the political process, leading to greater accountability from leaders.
The rise of democracy in Africa has had both positive and negative impacts, but overall it has contributed to the establishment of more accountable and transparent governance systems. With democratic institutions in place, African nations are better positioned to address the political and socio-economic challenges that they face.
$690 is invested in an account earning 2.2% interest (APR), compounded quarterly.
Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
Step-by-step explanation:
The formula to calculate the value of the account after t years, with principal P and annual percentage rate (APR) r compounded n times per year, is given by:
A = P(1 + r/n)^(nt)
In this case, P = $690, r = 0.022 (2.2% expressed as a decimal), n = 4 (compounded quarterly), and t is the number of years.
So the function to calculate the value of the account after t years is:
A(t) = 690(1 + 0.022/4)^(4t)
Simplifying and rounding to four decimal places, we get:
A(t) = 690(1.0055)^4t
To find the annual percentage yield (APY), we use the formula:
APY = (1 + r/n)^n - 1
In this case, r = 0.022 and n = 4, so:
APY = (1 + 0.022/4)^4 - 1
= 0.022321
Multiplying by 100 and rounding to two decimal places, we get an APY of 2.23%.
What rotation centered about the origin maps (4, − 7) to (7,4) ? 90° counterclockwise 180° counterclockwise 270° counterclockwise I don't know. ←
Answer:
What rotation centered about the origin maps (4, − 7) to (7,4) ? 90° counterclockwise 180° counterclockwise 270° counterclockwise I don't know. ←
Step-by-step explanation:
To map the point (4, -7) to (7, 4) by a rotation centered about the origin, we need to find the angle of rotation and direction.
We can start by finding the vector from the origin to (4, -7), which is <4, -7>. We want to rotate this vector to the vector from the origin to (7, 4), which is <7, 4>.
To do this, we need to find the angle between these two vectors. Using the dot product, we have:
<4, -7> · <7, 4> = (4)(7) + (-7)(4) = 0
Since the dot product is zero, we know that the two vectors are orthogonal, and the angle between them is 90 degrees.
To map (4, -7) to (7, 4) with a 90-degree rotation counterclockwise, we can use the matrix:
[0 -1]
[1 0]
Multiplying this matrix by the vector <4, -7>, we get:
[0 -1] [4] = [-7]
[1 0] [-7] [ 4]
which corresponds to the point (-7, 4). This matches our desired endpoint, so the answer is 90° counterclockwise.
Answer:
90° counterclockwise
Step-by-step explanation:
I am not sure if the picture helps or not. I am trying to show that I traced the point (4,-7). Then I have a plus sign at (0,0). I start rotating the tracing paper counterclockwise until I get to the point (7,4). I needed to turn one turn of the plus sign. That would be 90°
Helping in the name of Jesus.
consider the graph it f(x) = (1/2)^x
each graph shows the result of a transformation applied to function f
complete this statement given that g(x) = -f(x)
The graph of function g is graph _W,X,Y,Z_ because the graph of function g is the result of a ____vertical compression, vertical stretch, horizontal shift, reflection over the x axis____ applied to the graph of function f.
Answer:
graph Z , Horizontal Shift
Step-by-step explanation:
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What is the meaning of "The elements of F are all finite sequences (x1, x2, ..., xn) of elements of X "?
This is true, of course. The collection of any and all finite sequence of X's elements, along with the concatenation operation, is referred to as the free fuzzy set F over a set .
What does the math symbol X mean?The sentence xA denotes that x is an element of a set A since the symbol denotes set membership and meaning "is an element of". In those other words, x belongs to the group of (potentially many) items in set A.
What do math components consist of?
Components are also the components that constitute a set. A shared characteristic of the items can define a set. For instance, the set is the collection E of positive roughly equal numbers. Furthermore, F is a semigroup, since the operation of concatenation is associative: if and are elements of F, then Finally, F is a free semigroup over , which means that every element of F can be written uniquely as a product of elements of .
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Census data for a certain county shows that 19% of the adult residents are Hispanic. Suppose 92 people are called to jury duty and only 11 of them are Hispanic. Does this apparent underrepresentation of Hispanics call into question the fairness of the jury selection process? Again run a test using the PHANTOMS method to complete all parts of your problem.
Yes, the apparent under representation of Hispanics on the jury duty calls into question the fairness of the jury selection system, as the results of the hypothesis test suggest that the system may be systematically excluding Hispanics.
To determine whether the apparent underrepresentation of Hispanics on the jury selection system is statistically significant and calls into question its fairness, we need to perform a hypothesis test.
First, we set up the null hypothesis, which is the assumption that there is no difference between the proportion of Hispanics in the county population and the proportion of Hispanics in the jury pool. That is, the proportion of Hispanics in the jury pool is expected to be equal to 19%.
The alternative hypothesis is that there is a difference between the proportion of Hispanics in the county population and the proportion of Hispanics in the jury pool.
We can use a one-tailed z-test to test the null hypothesis, where the test statistic is calculated as:
z = (p - P) / sqrt(P(1-P)/n)
where p is the proportion of Hispanics in the jury pool, P is the proportion of Hispanics in the county population (0.19), and n is the sample size (72).
Plugging in the values, we get
z = (9/72 - 0.19) / sqrt(0.19*(1-0.19)/72) = -2.39
Assuming a significance level of 0.05, we compare the calculated z-value with the critical z-value of -1.645 (obtained from a standard normal distribution table). Since the calculated z-value is less than the critical z-value, we reject the null hypothesis and conclude that the proportion of Hispanics in the jury pool is significantly different from the proportion of Hispanics in the county population.
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evaluate cos2a if sin3a=2sina
Using triple angle formula the evaluation of the trigonometric identity cos(2a) are 1 and 1/2.
What is the value of cos2aWe can use the trigonometric identity cos(2a) = 1 - 2sin^2(a) to evaluate cos(2a), but first we need to find the value of sin(a) from the given equation.
Given: sin(3a) = 2sin(a)
We can expand sin(3a) using the triple angle formula:
[tex]sin(3a) = 3sin(a) - 4sin^3(a)[/tex]
Substituting the given equation into this, we get:
[tex]2sin(a) = 3sin(a) - 4sin^3(a)[/tex]
Simplifying, we can rearrange to get:
[tex]4sin^3(a) - sin(a) = 0[/tex]
Factorizing, we get:
[tex]sin(a)(4sin^2(a) - 1) = 0[/tex]
So, either sin(a) = 0 or 4sin^2(a) - 1 = 0.
If sin(a) = 0, then
[tex]cos(a) = \±1\\cos(2a) = cos^2(a) = 1.[/tex]
If 4sin^2(a) - 1 = 0, then we can solve for sin(a) to get:
[tex]sin(a) = \±\sqrt{(1/4)} = \±1/2[/tex]
If sin(a) = 1/2, then
[tex]cos(a) = \sqrt{(1 - sin^2(a))} = \sqrt{(1 - 1/4)} = \sqrt{3/2}[/tex]
Using the identity cos(2a) = 1 - 2sin^2(a), we can then calculate:
[tex]cos(2a) = 1 - 2sin^2(a) = 1 - 2(1/4) = 1/2[/tex]
If sin(a) = -1/2, then cos(a) = -√3/2, and using the same identity we get:
[tex]cos(2a) = 1 - 2sin^2(a) = 1 - 2(1/4) = 1/2[/tex]
So, we have two possible values for cos(2a): 1 and 1/2.
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